Question

Expand using Binomial Theorem .

Answer 1

The given expression ( 1+ x 2 − 2 x ) 4 is to be expanded.

Using Binomial theorem expansion,

( 1+ x 2 − 2 x ) 4 = C 4 0 ( 1+ x 2 ) 4 − C 4 1 ( 1+ x 2 ) 3 ( 2 x ) + C 4 2 ( 1+ x 2 ) 2 ( 2 x ) 2 − C 4 3 ( 1+ x 2 ) ( 2 x ) 3 + C 4 4 ( 2 x ) 4 = ( 1+ x 2 ) 4 −4 ( 1+ x 2 ) 3 ( 2 x )+6( 1+x+ x 2 4 )( 4 x 2 ) −4( 1+ x 2 )( 8 x 3 )+ 16 x 4 = ( 1+ x 2 ) 4 − 8 x ( 1+ x 2 ) 3 + 24 x 2 + 24 x +6− 32 x 3 − 16 x 2 + 16 x 4 = ( 1+ x 2 ) 4 − 8 x ( 1+ x 2 ) 3 +( 8 x 2 )+( 24 x )+6− 32 x 3 + 16 x 4 (1)

( 1+ x 2 ) 4 = C 4 0 ( 1 ) 4−0 + C 4 1 ( 1 ) 3 ( x 2 )+ C 4 2 ( 1 ) 2 ( x 2 ) 2 + C 4 3 ( 1 ) 3 ( x 2 ) 3 + C 4 4 ( x 2 ) 4 =1+4× x 2 +6× x 2 4 +4× x 3 8 + x 4 16 =1+2x+ 3 x 2 2 + x 3 2 + x 4 16 (2)

( 1+ x 2 ) 3 = C 3 0 ( 1 ) 3−0 + C 3 1 ( 1 ) 2 ( x 2 )+ C 3 2 ( 1 ) ( x 2 ) 2 + C 3 3 ( 1 ) ( x 2 ) 3 =1+ 3x 2 + 3 x 2 4 + x 3 8 (3)

From (1), (2) and (3), we get

( 1+ x 2 − 2 x ) 4 =1+2x+ 3 x 2 2 + x 3 2 + x 4 16 − 8 x ( 1+ 3x 2 + 3 x 2 4 + x 3 8 )+ 8 x 2 + 24 x +6− 32 x 3 + 16 x 4 =1+2x+ 3 x 2 2 + x 3 2 + x 4 16 − 8 x −12−6x− x 2 + 8 x 2 + 24 x +6− 32 x 3 + 16 x 4 = 16 x + 8 x 2 − 32 x 3 + 16 x 4 −4x+ x 2 2 + x 3 2 + x 4 16 −5 Thus, the expression ( 1+ x 2 − 2 x ) 4 has the expansion as 16 x + 8 x 2 − 32 x 3 + 16 x 4 −4x+ x 2 2 + x 3 2 + x 4 16 −5 .